In the chemical analysis of a rock the mass ratio of two radioactive isotopes is found to be `100:1`. The mean lives of the two isotopes are `4xx10^9` years and `2xx10^9` years, respectively. If it is assumed that at the time of formation the atoms of both the isotopes were in equal propotional, calculate the age of the rock. Ratio of the atomic weights of the two isotopes is `1.02:1`.

Explanation

Here, `(M_1)/(M_2)=100/1`
`tau_1=4xx10^9yrs tau_2=2xx10^9yrs`
Ratio of atomic weights, `(A_2)/(A_1)=1.02:1`
At `t=0, (N_0)_1=(N_0)_2`
If `N_1, N_2` are the number of atoms in the two isotopes at time t, therefore as
`M=NA or N=M/A`
`(N_1)/(N_2)=(M_1)/(M_2). (A_2)/(A_1)=100xx1/1.02 ....(i)`
for first isotope,
`N_1=(N_0)_1 e^(-lambda_1t)=(N_0)_1 e^(-t//tau_1)`
`N_2=(N_0)_2 e^(-lambda_2t)=(N_0)_2 e^(-t//tau_2)`
`(N_1)/(N_2)=e^(-t) (1/(tau_1)-1/(tau_2))=e^(t) (1/(tau_2)-1/(tau_1)).....(ii)`
form (i) and (ii)
`t (1/(tau_2)-1/(tau_1))=log_e(100/1.02)`
`t=log_e (100/1.02) .(tau_1tau_2)/(tau_1-tau_2)`
`=(2.3026(2-0.0086)xx4xx10^9xx2xx10^9)/((4xx10^9-2xx10^9))`
`t=1.834xx10^(10)years`
This is the age of the rock.